exp02 - Electronic Instrumentation Experiment 2 * Part A:...

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Electronic Instrumentation Experiment 2 * Part A: Intro to Transfer Functions and AC Sweeps * Part B: Phasors, Transfer Functions and Filters * Part C: Using Transfer Functions and RLC Circuits * Part D: Equivalent Impedance and DC Sweeps
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Part A Introduction to Transfer Functions and Phasors Complex Polar Coordinates Complex Impedance (Z) AC Sweeps
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Transfer Functions The transfer function describes the behavior of a circuit at V out for all possible V in . in out V V H
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Simple Example 3 2 1 3 2 * R R R R R V V in out + + + = k k k k k V V in out 3 2 1 3 2 * + + + = 6 5 = in out V V H V kt V t V then V kt V t V if out in 10 ) 2 2 sin( 5 ) ( 12 ) 2 2 sin( 6 ) ( + + = + + = π
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More Complicated Example What is H now? H now depends upon the input frequency ( ϖ = 2 π f) because the capacitor and inductor make the voltages change with the change in current.
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How do we model H? We want a way to combine the effect of the components in terms of their influence on the amplitude and the phase. We can only do this because the signals are sinusoids cycle in time derivatives and integrals are just phase shifts and amplitude changes
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We will define Phasors A phasor is a function of the amplitude and phase of a sinusoidal signal Phasors allow us to manipulate sinusoids in terms of amplitude and phase changes. Phasors are based on complex polar coordinates. Using phasors and complex numbers we will be able to find transfer functions for circuits. ) , ( φ A f V =
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Review of Polar Coordinates point P is at ( r p cos θ p , r p sin θ p ) 2 2 1 tan P P P P P P y x r x y + = = - θ
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Review of Complex Numbers z p is a single number represented by two numbers z p has a “real” part (x p ) and an “imaginary” part (y p ) j j j j j - = - = - 1 1 1
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Complex Polar Coordinates z = x+jy where x is A cos φ and y is A sin φ ϖ t cycles once around the origin once for each cycle of the sinusoidal wave ( ϖ =2 π f)
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Now we can define Phasors .) , ( sin cos , ) sin( ) cos( , ) cos( ) ( dropped is it so term each to common is t jA A V simply or t jA t A V let then t A t V if ϖ φ + = + + + = + = The real part is our signal. The two parts allow us to determine the influence of the phase and amplitude changes mathematically. After we manipulate the numbers, we discard the imaginary part.
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The “V=IR” of Phasors The influence of each component is given by Z, its complex impedance Once we have Z, we can use phasors to analyze circuits in much the same way that we analyze resistive circuits – except we will be using the complex polar representation. Z I V x x =
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Magnitude and Phase V of phase x y V V of magnitude A y x V jy x A j A V x x x x x φ = = = + + = + - 1 2 2 tan sin cos Phasors have a magnitude and a phase derived from polar coordinates rules.
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Influence of Resistor on Circuit Resistor modifies the amplitude of the signal by R Resistor has no effect on the phase R I V R R = ) sin( * ) ( ) sin( ) ( t A R t V then t A t I if R R ϖ = =
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Influence of Inductor on Circuit
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exp02 - Electronic Instrumentation Experiment 2 * Part A:...

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